Rive and Yoke

Rive went first. "Finding a number's factors," he told the children, "is finding every way it rives into equal rows. Thirty beasts — can we make equal pens? Try pens of two: fifteen pens of two, clean, no beast left over. So two is a factor. Try pens of three: ten pens of three, clean. Three is a factor. Try five: six pens of five. Five is a factor." He was thorough and slow, testing each number in turn, and the children began to see that a factor was simply a pen-size that used up the whole herd with nothing standing loose.

But it was slow work, testing every possible pen-size one after another, and a child near the back asked, plaintively, whether there was a way to know where to start instead of trying every number blindly.

Yoke, at the fence beside him, laughed. "There is," she said, "and it is my whole trade. Before you try to rive a number at all, ask the easiest question in the world: is it even? Pair the herd off — this beast with that, this with that. If every beast finds a partner, the number is even, and that means" — she paused, and looked at Rive, and something dawned on both of them at once — "that means two rives it cleanly. An even number can always be split into pens of two. So you never have to test two on an even number. You already know."

She showed them with the herd. Thirty beasts paired off with none left standing. "Even," she said. "So two is a factor — I didn't have to try it, I could see it. And an odd number? One beast always left standing, so pens of two never come out clean — two is not a factor, and you can skip straight past it without testing at all."

And then, at the fence, the two of them built the thing that was better than either craft alone — and they had the children do it, hands on the herd, to feel it work.

"Take thirty," Rive said, "and let's find its factors the fast way. Yoke — the easy question first." Yoke had the children pair the herd: even. "So two is in," said Rive, "and I can halve it — thirty splits into two fifteens, so now I only have to understand fifteen." He set fifteen beasts apart. "Yoke — is fifteen even?" The children paired them and found one left standing. "Odd," said Yoke. "So skip two entirely — don't waste a moment testing it." "Then I try three," said Rive, "and fifteen rives into three fives — and five and three are both stubborn, un-riveable primes, so we're done." The children looked at what they had built together: thirty, taken all the way apart into two, three, and five, and they had never once tested a pen-size blindly. Every hard split had been aimed by an easy question. Ask if it's even first. If yes, two's a factor — halve it. If no, skip two and move on. The pairing-test had become the first, fastest move of the factor-hunt.